Stratification and Arithmetic Dynamics on Character Varieties
Date
2022
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Abstract
If G is a reductive algebraic group over Z, the G-character variety of a finitely presented group F parameterizes the set of closed conjugation orbits in Hom(F,G). The group of automorphisms, Aut(F), acts on the representation variety, Hom(F,G), which leads to a natural action of the group of outer automorphisms, Out(F), on the character variety. In this thesis, we study the dynamics of the action of Out(F) on the finite field points of the character variety X_F(G). We provide a criterion in terms of subgroups of G for the action to be non-transitive on the non-trivial points of the representation variety and the character variety. We define free-type groups to be groups with elementary automorphisms similar to the Nielsen transformations of a free group. We then proceed to prove that the Aut(F) action is transitive on the set of epimorphisms from F to G when F is free-type. Additionally, we provide a characterization of free-type groups. Finally, we introduce the idea of asymptotic ratio as the ratio of the number of points in a maximal orbit to that in the variety as the order of the finite field goes to infinity. If the asymptotic ratio equals one, we say that the action is asymptotically transitive. We provide an upper bound for the asymptotic ratio in these cases and thus prove that the action is not asymptotically transitive on the SL_n- character varieties of Z^r for n=2,3.
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Keywords
Algebraic geometry and arithmetic dynamics, Asymptotic transitivity, Character varieties, E-polynomial, Free-type groups, Outer automorphism group action